Physics 207, Lecture 13, Oct. 15

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Physics 207, Lecture 13, Oct. 15

• Agenda Finish Chapter 10, start Chapter 11

• Chapter 10 Energy
• Potential Energy (gravity, springs)
• Kinetic energy
• Mechanical Energy
• Conservation of Energy
• Start Chapter 11, Work

• Assignment
• HW5 due tonight
• HW6 available today
• Monday, finish reading chapter 11

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Chapter 10 Energy

• Rearranging Newtons Laws gives (Fd vs. ½ mv2
  relationship)
• -2mg (yf yi ) m (vyf2 - vyi2 )
• or ? ½ m vyi2 mgyi ½ m vyf2 mgyf
• and adding ½ m vxi2 ½ m vzi2 and ½ m
  vxf2 ½ m vzf2
• ½ m vi2 mgyi ½ m vf2 mgyf
• where vi2 vxi2 vyi2 vzi2
• ½ m v2 terms are referred to as kinetic
  energy

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Energy

• If only conservative forces are present, the
  total energy (sum of potential, U, and kinetic
  energies, K) of a system is conserved.

K ½ mv2 U mgy
Ki Ui Kf Uf

• K and U may change, but E K Umech remains
  constant.

Emech is called mechanical energy
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Another example of a conservative system The
simple pendulum.

• Suppose we release a mass m from rest a distance
  h1 above its lowest possible point.
• What is the maximum speed of the mass and where
  does this happen ?
• To what height h2 does it rise on the other side ?

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Example The simple pendulum.

• What is the maximum speed of the mass and where
  does this happen ?
• E K U constant and so K is maximum when U
  is a minimum.

y
yh1
y0
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Example The simple pendulum.

• What is the maximum speed of the mass and where
  does this happen ?
• E K U constant and so K is maximum when U
  is a minimum
• E mgh1 at top
• E mgh1 ½ mv2 at bottom of the swing

y
yh1
h1
y0
v
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Example The simple pendulum.

• To what height h2 does it rise on the other
  side?
• E K U constant and so when U is maximum
  again (when K 0) it will be at its highest
  point.
• E mgh1 mgh2 or h1 h2

y
yh1h2
y0
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Lecture 13, Exercise 1Conservation of Mechanical
Energy
A block is shot up a frictionless 40 slope with
initial velocity v. It reaches a height h before
sliding back down. The same block is shot with
the same velocity up a frictionless 20
slope. On this slope, the block reaches height

1. 2h
2. h
3. h/2
4. Greater than h, but we cant predict an exact
   value.
5. Less than h, but we cant predict an exact value.

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Lecture 13, ExampleThe Loop-the-Loop again

• To complete the loop the loop, how high do we
  have to let the release the car?
• Condition for completing the loop the loop
  Circular motion at the top of the loop (ac v2 /
  R)
• Use fact that E U K constant !

Ubmgh
Recall that g is the source of the centripetal
acceleration and N just goes to zero is the
limiting case. Also recall the minimum speed at
the top is
Car has mass m
Umg2R
h ?
R
(A) 2R (B) 3R (C) 5/2 R (D) 23/2 R
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Lecture 13, ExampleThe Loop-the-Loop again

• Use E K U constant
• mgh 0 mg 2R ½ mv2
• mgh mg 2R ½ mgR 5/2 mgR
• h 5/2 R

h ?
R
(A) 2R (B) 3R 5/2 R (D) 23/2 R
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Lecture 13, ExampleSkateboard

• What speed will the skateboarder reach at bottom
  of the hill if there is no friction and the
  skeateboarder starts at rest?
• Assume we can treat the skateboarder as point
• Zero of gravitational potential energy is at
  bottom of the hill

m 25 kg
R5 m
R5 m
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Lecture 13, ExampleSkateboard

• What speed will the skateboarder reach at bottom
  of the hill if there is no friction and the
  skeateboarder starts at rest?
• Assume we can treat the skateboarder as point
• Zero of gravitational potential energy is at
  bottom of the hill

m 25 kg

• Use E K U constant
• Ebefore Eafter
• 0 mgR ½ mv2 0
• 2gR v2 ? v (2gR)½
• v (2 x 10 x 5)½ 10 m/s

R5 m
R5 m
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Potential Energy, Energy Transfer and Path

• A ball of mass m, initially at rest, is released
  and follows three difference paths. All surfaces
  are frictionless
• The ball is dropped
• The ball slides down a straight incline
• The ball slides down a curved incline
• After traveling a vertical distance h, how do the
  three speeds compare?

1
3
2
h
(A) 1 gt 2 gt 3 (B) 3 gt 2 gt 1 (C) 3 2 1
(D) Cant tell
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Lecture 13, Exercise 2 Potential Energy, Energy
Transfer and Path

• A ball of mass m, initially at rest, is released
  and follows three difference paths. All surfaces
  are frictionless
• The ball is dropped
• The ball slides down a straight incline
• The ball slides down a curved incline
• After traveling a vertical distance h, how do the
  speeds compare?

1. 1 gt 2 gt 3
2. 3 gt 2 gt 1
3. 3 2 1
4. Cant tell

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Potential Energy, Energy Transfer and Path

• A ball of mass m, initially at rest, is released
  and follows three difference paths. All surfaces
  are frictionless
• The ball is dropped
• The ball slides down a straight incline
• The ball slides down a curved incline
• After traveling a vertical distance h, how do the
  three speeds compare?

(A) 1 gt 2 gt 3 (B) 3 gt 2 gt 1 (C) 3 2 1
(D) Cant tell
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Elastic vs. Inelastic Collisions

• A collision is said to be elastic when energy as
  well as momentum is conserved before and after
  the collision.
  Kbefore Kafter
• Carts colliding with a perfect spring, billiard
  balls, etc.

• A collision is said to be inelastic when energy
  is not conserved before and after the collision,
  but momentum is conserved.
  
• Kbefore ? Kafter
• Car crashes, collisions where objects stick
  together, etc.

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Inelastic collision in 1-D Example 1

• A block of mass M is initially at rest on a
  frictionless horizontal surface. A bullet of
  mass m is fired at the block with a muzzle
  velocity (speed) v. The bullet lodges in the
  block, and the block ends up with a speed V.
• What is the initial energy of the system ?
• What is the final energy of the system ?
• Is energy conserved?

x
v
V
before
after
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Inelastic collision in 1-D Example 1

• What is the momentum of the bullet with speed v
  ?
• What is the initial energy of the system ?
• What is the final energy of the system ?
• Is momentum conserved (yes)?
• Is energy conserved? Examine Ebefore-Eafter

v
No!
V
x
before
after
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Example Fully Elastic Collision

• Suppose I have 2 identical bumper cars.
• One is motionless and the other is approaching it
  with velocity v1. If they collide elastically,
  what is the final velocity of each car ?
• Identical means m1 m2 m
• Initially vGreen v1 and vRed 0

• COM ? mv1 0 mv1f mv2f ? v1 v1f
  v2f
• COE ? ½ mv12 ½ mv1f2 ½ mv2f2 ? v12 v1f2
  v2f2
• v12 (v1f v2f)2 v1f2 2v1fv2f v2f2 ? 2
  v1f v2f 0
• Soln 1 v1f 0 and v2f v1 Soln 2 v2f
  0 and v1f v1

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Lecture 13, Exercise for homeElastic Collisions

• I have a line of 3 bumper cars all touching. A
  fourth car smashes into the others from behind.
  Is it possible to satisfy both conservation of
  energy and momentum if two cars are moving after
  the collision?
• All masses are identical, elastic collision.
• (A) Yes (B) No (C) Only in one special
  case

v
Before
v1
v2
After?
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Lecture 13, Exercise for homeElastic Collisions

• COM ? mv mv1 mv2 so v v1 v2
• COE ? ½ mv2 ½ mv12 ½ mv22
• v2 (v1 v2)2 v12 v22 ? v1 v2 0
• (A) Yes (B) No (C) Only in one special
  case

Before
After?
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Variable force devices Hookes Law Springs

• Springs are everywhere, (probe microscopes, DNA,
  an effective interaction between atoms)
• In this spring, the magnitude of the force
  increases as the spring is further compressed (a
  displacement).
• Hookes Law,
• Fs - k Dx
• Dx is the amount the spring is stretched or
  compressed from it resting position.

Rest or equilibrium position
F
Dx
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Hookes Law Spring

• For a spring we know that Fx -kx.

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Lecture 13, ExampleHookes Law

• Remembering Hookes Law, Fx -k Dx
• What are the units for the constant k ?
• (A) (B) (C) (D)

F is in kg m/s2 and dividing by m gives kg/s2 or
N/m
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Lecture 13, Exercise 2Hookes Law
(A) 50 N/m (B) 100 N/m (C) 400 N/m (D) 500 N/m
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Lecture 10, Exercise 2Hookes Law
8 m
9 m
What is the spring constant k ? SF 0 Fs
mg k Dx - mg Use k mg/Dx 5 N / 0.01 m
Fspring
50 kg
(A) 50 N/m (B) 100 N/m (C) 400 N/m (D) 500 N/m
mg
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F-x relation for a foot arch
Force (N)
Displacement (mm)
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F-x relation for a single DNA molecule
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Measurement technique optical tweezers
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Lecture 13, Oct. 15

• Chapter 10 Energy
• Potential Energy (gravity, springs)
• Kinetic energy
• Mechanical Energy
• Conservation of Energy
• Chapter 11, Work

• Assignment
• HW5 due tonight
• HW6 available today
• Monday, finish Chapter 11